Wednesday, January 15, 2020 - 14:00 in V3-201
Regularisation and Long-Time Behaviour of Random Systems: Tamed MHD Equations and Random Attractors for Locally Monotone SPDEs
A talk in the Bielefeld Stochastic Afternoon series by
Andre Schenke from Bielefeld
| Abstract: |
In this talk, I will give an overview of the results of my PhD
thesis. The first part is devoted to a new model for the flow of an
electrically conducting fluid through a porous medium, the \emph{tamed
magnetohydrodynamics (TMHD) equations}. We will discuss regularisation
schemes and motivation for fluid dynamical equations, and prove
existence, uniqueness and regularity in the deterministic case.
Furthermore, we can show that as the taming parameter tends to infinity,
the solution to the TMHD equations converges in a suitable sense to a
weak solutino of the MHD equations.
In the stochastic case, we show existence and uniqueness for the whole
space and the torus, as well as existence of a Feller semigroup and an
invariant measure for the latter case.
The second part (joint work with B. Gess and W. Liu) deals with the
long-time behaviour of solutions to SPDEs with locally monotone
coefficients with additive L\'{e}vy noise. Under quite general
assumptions, we prove existence of a random dynamical system as well as
a random attractor. This serves as a unifying framework for a large
class of examples. We will briefly discuss two of these: stochastic 2D
Navier-Stokes equations and the Leray-$\alpha$ model of fluid dynamics.
|
Back
